Work

Work:

In order to do work, a force must be applied and there must be some movement or displacement in the direction of that force. The work done by force acting on an object is equal to the magnitude of the force multiplied by the distance traveled in the direction of the force. The work only has size, not direction. Therefore, work is a scalar quantity.

 Work formula

 The work done by a force is defined as the product of the component of the force in the direction of displacement and the magnitude of that displacement. 

 W is the work done, F is the force, d is the displacement, θ is the angle between the force and the displacement, and F cosθ is the component of the force in the direction of the displacement.

 The work equation tells us that no matter how great the force is, if there is no displacement,  no work is done. In summary, we can say that no work is done if:

· displacement is zero

· Power is zero

·  Force and distance are orthogonal.

 Work unit

 The SI unit of work is the joule (J). For example, if you apply a force of 5 Newton’s to an object and it moves 2 meters, the work is 10 Newton meters or 10 joules. Note that 1 J = 1 N ⋅ m = 1 kg ⋅ m2/s2.

 An example of work

 An object is pulled horizontally across the surface by a force of 100 N acting parallel to the surface. Find the amount of work done by the force in moving the object a distance of 8 m.  Solution:

 

 Considering:  F = 100 N, d = 8 m

 Since F and d are in the same direction, θ = 0, [θ is the angle of the force with respect to the direction of motion],

 W = F d Cos θ

 L = 100 x 8 x Cos 0

 W = 800 J [from Cos 0 = 1]

Derivation of work energy theorem

 This is the derivation of the work-energy theorem. The work-energy theorem, also known as the principle of work and kinetic energy, states that the total work done by the sum of all the forces acting on a particle is equal to the change in the kinetic energy of the particle. This explanation can be extended to rigid bodies by describing the effects of rotational kinetic energy and torque.

 Kinetic Energy

 Kinetic energy is the energy that an object possesses due to its motion or motion. Simply put, there is kinetic energy that can be viewed as the motion of an object, particle, or group of particles.

Derivation of work-energy theorem

 The work 'W' done by the net force acting on the particle is equal to the change in kinetic energy (KE) of the particle.


 Consider the case where the resultant force "F" is constant in both direction and magnitude and is parallel to the velocity of the particle. A particle moves along a straight line with constant acceleration. The relationship between  acceleration and  net force is given by the equation "F = ma" (Newton's second law of motion) and the  displacement "d" of the particle can be determined by the equation:

 The work of the net force is calculated as the product of its magnitude (F=ma) and the displacement of the particle. Substituting the above equations gives:



Work done with variable forces

 What work does a variable force do? It is interesting to know that the forces we encounter every day are mostly of a variable nature, which is defined as a changing force. A force is said to act on a system if, when the force is applied, motion occurs in the system in the direction of the force. In the case of a variable force, integration is necessary to calculate the work done.

 As we know, the work done by a constant force F moving an object by Δx can be expressed as :

 

 W = F.Δx

For a variable force, the work is calculated by integration. For example, in the case of a spring, the force acting on any object attached to a horizontal spring can be given as:

 Fs = -kx

  where,

  k is the spring constant

 x is the displacement of the fixed object 

 We can see that this force is proportional to the displacement of the object from its equilibrium position, so the force acting on each compression and extension of the spring is different. So the infinitesimally small contributions of work done at each moment must be added to calculate the total amount of work done. 

 The integral is evaluated as follows:


 Work done by a variable force

Force transition diagram


 A graphical interpretation of the force in this force-displacement diagram helps us understand this concept more clearly. Force transition diagram

 

 Consider the variable force versus displacement curve as shown in the figure. Here, the small divisions represent the displacement Δx due to the force F(x) acting on that point. Assuming that the quantity Δx is small,  the force F(x) acting during that time can be considered a constant force. The area enclosed by a rectangle with a length equal to the magnitude of the force F(x) and a width equal to the displacement Δx gives the work done by the force during its duration. 

 Mathematically, ΔW =F (x) Δx

  By adding successive rectangles, the total work done can be written as:

  Work done by a variable force


 Assuming that the displacements approach zero, the following equation gives the total work done by the force.

  Work done by a variable force


 Thus, the work done by a variable force can be expressed as the definite integral of the force of any system with respect to the displacement